The Planar Coleman--Gurtin model with Beltrami conductivity
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911492684644352 |
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| author | Di Plinio, Francesco |
| author_facet | Di Plinio, Francesco |
| contents | This article addresses the planar Coleman--Gurtin heat equation with memory on a bounded domain, with rough anisotropic diffusion $A_μ$, typical of heterogeneous or composite media and encoded by a Beltrami coefficient $μ\in L^\infty(Ω)$ satisfying $\|μ\|_{\infty}<1$.
First, under no additional smoothness assumptions on $μ$, solutions with $H^1_0(Ω)$-based initial data enter a time-averaged $L^\infty(Ω)$ regime, and instantaneously regularize into the second-order graph space $D(A_μ)$.
Assuming in addition $μ\in W^{1,2}(Ω)$, this regularization upgrades to $W^{2,p}(Ω)$ for every $1<p<2$, and we construct regular global and exponential attractors of finite fractal dimension, for both the $L^2(Ω)$ and $H^1_0(Ω)$-based dynamics. The proof combines the instantaneous smoothing method of Chekroun, Di Plinio, Glatt-Holtz and Pata with maximal parabolic regularity for divergence-form operators with measurable coefficients, and with planar quasiconformal Beltrami estimates recently obtained in work by Green, Wick and the author. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_05983 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Planar Coleman--Gurtin model with Beltrami conductivity Di Plinio, Francesco Analysis of PDEs Classical Analysis and ODEs Complex Variables Primary: 35B41, 35K05, 30C62. Secondary: 45K05, 47H20 This article addresses the planar Coleman--Gurtin heat equation with memory on a bounded domain, with rough anisotropic diffusion $A_μ$, typical of heterogeneous or composite media and encoded by a Beltrami coefficient $μ\in L^\infty(Ω)$ satisfying $\|μ\|_{\infty}<1$. First, under no additional smoothness assumptions on $μ$, solutions with $H^1_0(Ω)$-based initial data enter a time-averaged $L^\infty(Ω)$ regime, and instantaneously regularize into the second-order graph space $D(A_μ)$. Assuming in addition $μ\in W^{1,2}(Ω)$, this regularization upgrades to $W^{2,p}(Ω)$ for every $1<p<2$, and we construct regular global and exponential attractors of finite fractal dimension, for both the $L^2(Ω)$ and $H^1_0(Ω)$-based dynamics. The proof combines the instantaneous smoothing method of Chekroun, Di Plinio, Glatt-Holtz and Pata with maximal parabolic regularity for divergence-form operators with measurable coefficients, and with planar quasiconformal Beltrami estimates recently obtained in work by Green, Wick and the author. |
| title | The Planar Coleman--Gurtin model with Beltrami conductivity |
| topic | Analysis of PDEs Classical Analysis and ODEs Complex Variables Primary: 35B41, 35K05, 30C62. Secondary: 45K05, 47H20 |
| url | https://arxiv.org/abs/2603.05983 |